Rhombic Dodecahedral Number

A figurate number which is constructed as a centered cube with a square pyramid appended to each face,

			(1)
			(2)

where  is a centered cube number and  is a square pyramidal number. The first few are 1, 15, 65, 175, 369, 671, ... (OEIS A005917). The generating function of the rhombic dodecahedral numbers is

(3)

A related set of numbers is the number of cubes in the Haűy construction of the rhombic dodecahedron, given by

(4)

for  an odd number. Re-indexing with  then gives

(5)

giving the first few values 1, 33, 185, 553, 1233, ... (OEIS A046142).

REFERENCES:

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53-54, 1996.

Sloane, N. J. A. Sequences A005917/M4968 and A046142 in "The On-Line Encyclopedia of Integer Sequences."